Rules interchanged using the Rule Interchange Format RIF may depend on or be used in combination with RDF data and RDF Schema or OWL ontologies. This document, developed by the Rule Interchange Format (RIF) Working Group, specifies the interoperation between RIF and the data and ontology languages RDF, RDF Schema, and OWL.

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1 Overview of RDF and OWL Compatibility

The Rule Interchange Format (RIF) is a format for interchanging rules over the Web. Rules that are exchanged using RIF may refer to external data sources and may be based on data models that are represented using a language different from RIF. The Resource Description Framework RDF [RDF-Concepts] is a Web-based language for the representation and exchange of data; RDF Schema (RDFS) [RDF-Schema] and the OWL Web Ontology Language [OWL2-Syntax] are Web-based languages for representing and exchanging ontologies. This document specifies how combinations of RIF documents and RDF data and RDFS and OWL ontologies are interpreted; i.e., it specifies how RIF interoperates with RDF, RDFS, and OWL. We consider here OWL 2 [OWL2-Syntax], which is an extension of OWL 1 [OWL-Reference]. Therefore, the notions defined in this document also apply to combinations of RIF documents with OWL 1 ontologies.

We consider here the RIF Basic Logic Dialect (BLD) [RIF-BLD] and RIF Core [RIF-Core], a subset of RIF BLD.
The RIF Production Rule Dialect (PRD) [RIF-PRD] is an extension of RIF Core. Interoperability between RIF and RDF/OWL is only defined for the Core subset of PRD. In the remainder, when speaking about RIF documents and rules, we refer to RIF Core and BLD.

RDF data and RDFS and OWL ontologies can be represented using RDF graphs. There exist several alternative syntaxes for OWL ontologies; however, for exchange purposes it is assumed they are represented using RDF graphs.

Several syntaxes have been proposed for the exchange of RDF graphs, the normative syntax being RDF/XML [RDF-Syntax]. RIF does not provide a format for exchanging RDF graphs; it is assumed that RDF graphs are exchanged using RDF/XML, or any other syntax that can be used for representing or exchanging RDF graphs.

A typical scenario for the use of RIF with RDF/OWL is the exchange of rules that use RDF data and/or RDFS or OWL ontologies: an interchange partner A has a rules language that is RDF/OWL-aware, i.e., it supports the use of RDF data, it uses an RDFS or OWL ontology, or it extends RDF(S)/OWL. A sends its rules using RIF, possibly with references to the appropriate RDF graph(s), to partner B. B receives the rules and retrieves the referenced RDF graph(s). The rules are translated to the internal rules language of B and are processed, together with the RDF graphs, using the RDF/OWL-aware rule engine of B. The use case Vocabulary Mapping for Data Integration [RIF-UCR] is an example of the interchange of RIF rules that use RDF data and RDFS ontologies.

A specialization of this scenario is the publication of RIF rules that refer to RDF graphs; publication is a special kind of interchange: one to many, rather than one-to-one. When a rule publisher A publishes its rules on the Web, there may be several consumers that retrieve the RIF rules and RDF graphs from the Web, translate the RIF rules to their respective rules languages, and process them together with the RDF graphs in their own rules engines. The use case Publishing Rules for Interlinked Metadata [RIF-UCR] illustrates the publication scenario.

Another specialization of the exchange scenario is the Interchange of Rule Extensions to OWL [RIF-UCR]. The intention of the rule publisher in this scenario is to extend an OWL ontology with rules: interchange partner A has a rules language that extends OWL. A splits its ontology+rules description into a separate OWL ontology and a RIF document, publishes the OWL ontology, and sends (or publishes) the RIF document, which includes a reference to the OWL ontology. A consumer of the rules retrieves the OWL ontology and translates the ontology and document into a combined ontology+rules description in its own rule extension of OWL.

A RIF document that refers to (imports) RDF graphs and/or RDFS/OWL ontologies, or any use of a RIF document with RDF graphs, is viewed as a combination of a document and a number of graphs and ontologies. This document specifies how, in such a combination, the document and the graphs and ontologies interoperate in a technical sense, i.e., the conditions under which the combination is satisfiable (i.e., consistent), as well as the entailments (i.e., logical consequences) of the combination.
The interaction between RIF and RDF/OWL is realized by connecting the model theory of RIF [RIF-BLD] with the model theories of RDF [RDF-Semantics] and OWL [OWL2-Semantics], respectively.

The notation of certain symbols in RIF, particularly IRIs and plain literals, is slightly different from the notation in RDF/OWL. These differences are illustrated in the Section Symbols in RIF Versus RDF/OWL.

The RDF semantics specification [RDF-Semantics] defines four normative notions of entailment for RDF graphs: Simple, RDF, RDFS, and Datatype entailment. OWL 2 specifies two different semantics, with corresponding notions of entailment: the direct semantics [OWL2-Semantics], which specifies OWL 2 DL entailment, and the RDF-based semantics [OWL2-RDF-Based-Semantics], which specifies OWL 2 Full entailment. This document specifies the interaction between RIF and RDF/OWL for all six notions. The Section RDF Compatibility is concerned with the combination of RIF and RDF/RDFS. The combination of RIF and OWL is addressed in the Section OWL Compatibility. The semantics of the interaction between RIF and OWL 2 DL is close in spirit to [SWRL].

RIF provides a mechanism for referring to (importing) RDF graphs and a means for specifying the profile of this import, which corresponds to the intended entailment regime. The Section Importing RDF and OWL in RIF specifies how such import statements are used for representing RIF-RDF and RIF-OWL combinations.

The Appendix: Embeddings (Informative) describes how reasoning with combinations of RIF rules with RDF and OWL 2 RL (a subset of OWL 2 DL) can be reduced to reasoning with RIF documents. This reduction can be seen as an implementation hint for interchange partners who do not have RDF/OWL-aware rule systems, but want to process RIF rules that import RDF graphs and OWL ontologies. In terms of the aforementioned scenario: if the interchange partner B does not have an RDF/OWL-aware rule system, but B can process RIF rules, then the appendix explains how the rule system of B could be used for processing RIF-RDF/OWL combinations.

Throughout this document the following conventions are used when writing RIF and RDF statements in examples and definitions.

RDF triples are written using the Turtle syntax [Turtle]: triples are written as s p o, where s, p, and o are blank nodes _:x, IRIs delimited with '<' and '>', compact IRIs prefix:localname, plain literals without language tags "literal", plain literals with language tags "literal"@lang, or typed literals "literal"^^datatype-IRI.

The following namespace prefixes are used throughout this document: ex refers to http://example.org/example#, xs refers to http://www.w3.org/2001/XMLSchema#, rdf refers to http://www.w3.org/1999/02/22-rdf-syntax-ns#, rdfs refers to http://www.w3.org/2000/01/rdf-schema#, owl refers to http://www.w3.org/2002/07/owl#, and rif refers to http://www.w3.org/2007/rif#.

Symbol spaces can be seen as groups of constants. Every datatype is a symbol space, but there are symbol spaces that are not datatypes. For example, the symbol space rif:iri groups all IRIs.
The correspondence between constant symbols in RDF graphs and RIF documents is explained in Table 1.

IRIs, i.e., constants of the form "IRI"^^rif:iri, may be written as <IRI> or as compact IRIs [CURIE], i.e., as prefix:localname, where prefix is understood to refer to an IRI namespace-IRI, and prefix:localname stands for the IRI (IRI) obtained by concatenating namespace-IRI and localname.

Strings, i.e., constants of the form "my string"^^xs:string may be written as "my string".

RIF does not have a notion corresponding exactly to RDF blank nodes. RIF local symbols, written _symbolname, have some commonality with blank nodes; like the blank node label, the name of a local symbol is not exposed outside of the document. However, in contrast to blank nodes, which are essentially existentially quantified variables, RIF local symbols are constant symbols. In many applications and deployment scenarios, this difference may be inconsequential. However the results will differ when such symbols are used in a non-assertional context, such as in a query pattern or rule body.

Finally, variables in the bodies of RIF rules or in query patterns may be existentially quantified, and are thus similar to blank nodes; however, RIF BLD does not allow existentially quantified variables to occur in rule heads.

3 RDF Compatibility

This section specifies how a RIF document interacts with a set of RDF graphs in a RIF-RDF combination. In other words, how rules can "access" data in the RDF graphs.

There is a correspondence between statements in RDF graphs and certain kinds of formulas in RIF. Namely, there is a correspondence between RDF triples of the form s p o and RIF frame formulas of the form s'[p' -> o'], where s', p', and o' are RIF symbols corresponding to the RDF symbols s, p, and o, respectively. This means that whenever a triple s p o is satisfied, the corresponding RIF frame formula s'[p' -> o'] is satisfied, and vice versa.

Consider, for example, a combination of an RDF graph that contains the triples

ex:john ex:brotherOf ex:jack .
ex:jack ex:parentOf ex:mary .

saying that ex:john is a brother of ex:jack and ex:jack is a parent of ex:mary,
and a RIF document that contains the rule

which says that whenever some x is a brother of some y and y is a parent of some z, then x is an uncle of z. From this combination the RIF frame formula :john[:uncleOf -> :mary], as well as the RDF triple :john :uncleOf :mary, are consequences of this combination.

Note that blank nodes cannot be referenced directly from RIF rules, since blank nodes are local to a specific RDF graph. Variables in RIF rules do, however, range over objects denoted by blank nodes. So, it is possible to "access" an object denoted by a blank node from a RIF rule using a variable in a rule.

The following example illustrates the interaction between RDF and RIF in the face of blank nodes.

Consider a combination of an RDF graph that contains the triple

_:x ex:hasName "John" .

saying that there is something, denoted here by a blank node, which has the name "John", and a RIF document that contains the rules

which says that whenever there is some x that has some name y, then x is of type ex:named and http://a has a property http://p with value y.

From this combination the following RIF condition formulas can be derived:

Exists ?z (?z[rdf:type -> ex:named])
<http://a>[<http://p> -> "John"]

as can the following RDF triples:

_:y rdf:type ex:named .
<http://a> <http://p> "John" .

However, there is no RIF constant symbol t such that t[rdf:type -> ex:named] can be derived, because there is no constant that represents the named individual.

Note that, even when considering Simple entailment, not every combination is satisfiable. In fact, not every RIF document has a model. For example, the RIF BLD document consisting of the fact

"a"="b"

does not have a model, since the symbols "a" and "b" are mapped to the (distinct) character strings "a" and "b", respectively, in every semantic structure.

One consequence of the difference of the alphabets of RDF and RIF is that IRIs of the form http://iri and typed literals of the form "http://iri"^^rif:iri that occur in an RDF graph are treated the same in RIF-RDF combinations, even if the RIF document is empty. However, documents importing RDF graphs containing typed literals of the form "http://iri"^^rif:iri must be rejected.

Plain literals without language tags of the form "mystring" and typed literals of the form "mystring"^^xs:string also correspond. For example, consider the combination of an empty document and an RDF graph that contains the triple

<http://a> <http://p> "abc" .

This combination entails, among other things, the following frame formula:

<http://a>[<http://p> -> "abc"^^xs:string]

as well as the following triple:

<http://a> <http://p> "abc"^^xs:string .

These entailments are sanctioned by the semantics of plain literals and xs:strings.

Lists in RDF (also called collections) have a natural correspondence to RIF lists. For example, the RDF list _:l1 rdf:first ex:b . _:l1 rdf:rest rdf:nil . corresponds to the RIF list List(ex:b). And so, the combination of the empty RIF document with the RDF graph

Editor's Note: This last example shows there is a 1-to-1 correspondence between RIF and RDF lists. Note that 1-to-1 correspondence (condition 10 in common-RIF-RDF-interpetations) is at risk. The previous example shows that RIF lists extend RDF lists (condition 9 in common-RIF-RDF-interpetations); this feature is not at risk.

The remainder of this section formally defines combinations of RIF rules with RDF graphs and the semantics of such combinations. A combination consists of a RIF document and a set of RDF graphs. The semantics of combinations is defined in terms of combined models, which are pairs of RIF and RDF interpretations. The interaction between the two interpretations is defined through a number of conditions. Entailment is defined as model inclusion, as usual.

3.1 Syntax of RIF-RDF Combinations

This section first reviews the definitions of RDF vocabularies and RDF graphs, after which RIF-RDF combinations are formally defined. The section concludes with a review of definitions related to datatypes and typed literals.

When clear from the context, RIF-RDF combinations are referred to simply as combinations.

3.1.3 Datatypes and Typed Literals

Even though RDF allows the use of arbitrary datatype IRIs in typed literals, not all such datatype IRIs are recognized in the semantics. In fact, Simple entailment does not recognize any datatype and RDF and RDFS entailment recognize only the datatype rdf:XMLLiteral. To facilitate discussing datatypes, and specifically datatypes supported in specific contexts (required for D-entailment),RIF-D-entailment), the notion of datatype maps [RDF-Semantics] is used.

RDFS, specifically D-entailment,RIF-D-entailment, allows the use of arbitrary datatype maps, as long as rdf:XMLLiteral is in the domain of the map. RIF BLD requires a number of additional datatypes to be included; these are the RIF-required datatypes [RIF-DTB].

When checking consistency of a combination < R,S> or entailment of a graph S or RIF formula φ by a combination < R,S>, the set of considered datatypes is the union of the set of RIF-required datatypes and the sets of datatypes referenced in R, the documents imported intoR, and φ (when considering entailment of φ).

Definition. Let DTS be a set of datatypes. A datatype map D is conforming with DTS if it satisfies the following conditions:

Every IRI identifying a datatypes in DTS is in the domain of D.

D maps each IRI in its domain to the datatype in identified by that IRI in T. ☐

Note that it follows from the definition that every datatype used in the RIF document in the combination or the entailed RIF formula (when considering entailment questions) is included in any datatype map conforming to the set of considered datatypes. There may be datatypes used in an RDF graph in the combination that are not included in such a datatype map.

Definition. Given a datatype map D, a typed literal (s, d) is a well-typed literal if

d is in the domain of D and s is in the lexical space of D(d) or

d is the IRI of a symbol space required by RIF BLD and s is in the lexical space of the symbol space. ☐

3.2 Semantics of RIF-RDF Combinations

The semantics of RIF-RDF combinations is defined through a combination of the RIF and RDF model theories, using a notion of common models. These models are then used to define satisfiability and entailment in the usual way. Combined entailment extends both entailment in RIF and entailment in RDF.

The RDF Semantics document [RDF-Semantics] defines four normative kinds of interpretations, as well as corresponding notions of satisfiability and entailment:

Those four types of interpretations are reflected in the definitions of satisfaction and entailment in this section.

3.2.1 Interpretations

This section defines the notion of common-RIF-RDF-interpretation, which is an interpretation of a RIF-RDF combination. This common-RIF-RDF-interpretation is the basis for the definitions of satisfaction and entailment in the following sections.

The correspondence between RIF semantic structures (interpretations) and RDF interpretations is defined through a number of conditions that ensure the correspondence in the interpretation of names (i.e., IRIs and literals) and formulas, i.e., the correspondence between RDF triples of the form s p o and RIF frames of the form s'[p' -> o'], where s', p', and o' are RIF symbols corresponding to the RDF symbols s, p, and o, respectively (cf. the Section Symbols in RIF Versus RDF/OWL).

As defined in [RIF-BLD], a semantic structureI is a tuple of the form <TV, DTS, D, Dind,
Dfunc, IC, IV,
IF, INF, Ilist, Itail, Iframe,
Isub, Iisa, I=,
Iexternal, Itruth>. The specification of RIF-RDF compatibility is only concerned with DTS, D, IC, IV, Ilist, Itail, Iframe, Isub, Iisa, and Itruth. The other mappings that are parts of a semantic structure are not used in the definition of combinations.

DTS is the set of datatypes, which have associated datatype identifiers,

D is a set (the domain),

Dind is a non-empty subset of D,

Dfunc is a non-empty subset of D,

IC is a mapping from constants to D such that constants in individual position are mapped to Dind and constants in function positions are mapped to Dfunc,

IV is a mapping from Var to Dind,

Ilist is an injective mapping from Dind* to Dind,

Itail is a mapping from Dind+×Dind to Dind,

Iframe is a mapping from Dind to functions of the form SetOfFiniteBags(Dind × Dind) → D,

Isub is a mapping from Dind × Dind to D,

Iisa is a mapping from Dind × Dind to D, and

Itruth is a mapping from D to TV.

For the purpose of the interpretation of imported documents, RIF BLD defines the notion of semantic multi-structures, which are nonempty sets of semantic structures of the form {J,I; Ii1, Ii2, ...} that differ only in interpretation of local constants. The structure I in the above is used to interpret document formulas, and will be used to specify RIF combinations.

Note that, if n > 0, there may be several lists referred to by l1, since there is no restriction, in general, on the rdf:first elements and the rdf:rest successors.

3.2.1.3 Common RIF-RDF Interpretations

Definition. A common-RIF-RDF-interpretation is a pair (Î, I), where Î is a semantic multi-structure of the form {J,I; Ii1, Ii2, ...}, and I is anRDFaSimple interpretation of a vocabulary V, such that the following conditions hold:

(IR union IP) = Dind;

IP is a superset of the set of all k in Dind such that there exist some a, b in Dind and Itruth(Iframe(a)(k,b))=t;

Condition 1 ensures that the combination of resources and properties corresponds exactly to the RIF domain; note that if I is an RDF-, RDFS-, or D-interpretation, IP is a subset of IR, and thus IR=Dind. Condition 2 ensures that the set of RDF properties at least includes all elements that are used as properties in frames in the RIF domain. Condition 3 ensures that all concrete values in Dind are included in LV (by definition, the value spaces of all considered datatypes are included in Dind). Condition 4 ensures that RDF triples are interpreted in the same way as frame formulas. Condition 5 ensures that IRIs are interpreted in the same way. Condition 6 ensures that typed literals are interpreted in the same way. Note that no correspondences are defined for the mapping of names in RDF that are not symbols of RIF, e.g., ill-typed literals and RDF URI references that are not absolute IRIs. Condition 7 ensures that typing in RDF and typing in RIF correspond, i.e., a rdf:type b is true iff a # b is true. Condition 8 ensures that whenever a RIF subclass statement holds, the corresponding RDF subclass statement holds as well, i.e., a rdfs:subClassOf b is true if a ## b is true. Finally, condition 9 requires the existence of an RIF list for every RDF list and condition 10 in addition requires the existence of an RDF list for every RIF list.

Condition 10 (which implies condition 9), which ensures a one-to-one correspondence between RDF and RIF lists, is AT RISK and may be removed based on implementation experience.

3.2.2 Satisfaction and Models

The notion of satisfiability refers to the conditions under which a common-RIF-RDF-interpretation (Î, I) is a model of a combination < R, S>. The notion of satisfiability is defined for all four entailment regimes of RDF (i.e., Simple, RDF, RDFS, and D). The definitions are all analogous. Intuitively, a common-RIF-RDF-interpretation (Î, I) satisfies a combination < R, S> if Î is a model of R and I satisfies S. Formally:

Rdf-, RDFS-, and D-satisfiabilityRIF-D-satisfiability are defined through additional restrictions on I:

Definition. A model (Î, I) of a combination C is an RDF-modelRIF-RDF-model of C if I is an RDF-interpretation; in this case C is RDF-satisfiableRIF-RDF-satisfiable.

A model (Î, I) of a combination C is an RDFS-modelRIF-RDFS-model of C if I is an RDFS-interpretation; in this case C is RDFS-satisfiableRIF-RDFS-satisfiable.

Let (Î, I) be a model of a combination C and let D be a datatype map conforming with the set DTS of datatypes in I. (Î, I) is a D-modelRIF-D-model of C if I is a D-interpretation; in this case C is D-satisfiableRIF-D-satisfiable. ☐

3.2.3 Entailment

Using the notions of models defined above, entailment is defined in the usual way, i.e., through inclusion of sets of models.

Note that simple entailment in combination with an empty ruleset is not the same as simple entailment in RDF, since certain entailments involving datatypes are enforced by the RIF semantics in combinations, cf. the example involving strings and plain literals above.

4 OWL Compatibility

This section specifies how a RIF document interacts with a set
of OWL ontologies in a RIF-OWL combination. The semantics of
combinations is defined for OWL 2
[OWL2-Syntax]. Since OWL
2 is an extension of OWL 1
[OWL-Reference], the
specification in this section applies also to combinations of
RIF documents with OWL 1 ontologies.

OWL 2 specifies two different variants of the language, each
having its own semantics, namely OWL 2 DL
[OWL2-Syntax] (called
simply OWL 2 in the specification) and OWL 2 Full
[OWL2-RDF-Based-Semantics].
OWL 1 Lite and
OWL 1 DL [OWL-Reference],
which are sublanguages of OWL 1, can be seen as syntactical
subsets of OWL 2 DL. The key difference between OWL 2 DL and
OWL 2 Full is that the semantics of OWL 2 DL
[OWL2-Semantics] is
based on standard Description Logic semantics, whereas the
semantics of OWL 2 Full is an extension of the RDFS semantics.

The syntax of OWL 2 is defined in terms of a structural
specification, and there is a mapping to an RDF representation
for interchange. The RDF representation of OWL 2 DL does not
extend the RDF syntax, but rather restricts it: every OWL 2 DL
ontology in RDF form is an RDF graph, but not every RDF graph
is an OWL 2 DL ontology. OWL 2 Full and RDF have the same
syntax: every RDF graph is an OWL 2 Full ontology and vice
versa. This syntactical difference is reflected in the
definition of RIF-OWL compatibility: combinations of RIF with
OWL 2 DL are based on the OWL 2 structural model, whereas
combinations with OWL 2 Full are based on the RDF syntax.

Since the OWL 2 Full syntax is the same as the RDF syntax and
the OWL 2 Full semantics is an extension of the RDF semantics,
the definition of RIF-OWL 2 Full compatibility is an extension
of RIF-RDF compatibility. However, defining RIF-OWL DL
compatibility in the same way would entail losing certain
semantic properties of OWL 2 DL. One of the main reasons for
this is the difference in the way classes and properties are
interpreted in OWL 2 Full and OWL 2 DL. In the Full variant,
classes and properties are interpreted as objects in the domain
of interpretation, which are then associated with subsets of,
respectively binary relations over the domain of
interpretation, using the rdf:type property and the
extension function IEXT, as in RDF. In the DL variant, classes
and properties are directly interpreted as subsets of,
respectively binary relations over the domain. This is a key
property of the first-order logic nature of Description Logic
semantics and enables the use of Description Logic reasoning
techniques for processing OWL 2 DL descriptions. Defining
RIF-OWL DL compatibility as an extension of RIF-RDF
compatibility would define a correspondence between OWL 2 DL
statements and RIF frame formulas. Since RIF frame formulas
are interpreted using an extension function, as in RDF,
defining the correspondence between them and OWL 2 DL
statements would change the semantics of OWL statements, even
if the RIF document were empty.

A RIF-OWL combination that is faithful to the first-order
nature of the OWL 2 DL semantics requires interpreting classes
and properties as sets and binary relations, respectively,
suggesting that a correspondence could be defined with unary
and binary predicates. It is, however, also desirable that
there be uniform syntax for the RIF component of both RIF-OWL 2 DL
and RIF-RDF/OWL 2 Full combinations, because one may not know at
the time of constructing the rules which type of inference will
be used. Consider, for example, an RDF graph S consisting of
the following statements

_:x rdf:type owl:Ontology .
a rdf:type C .

and a RIF document with the rule

Forall ?x (?x[rdf:type -> D] :- ?x[rdf:type -> C])

The combination of the two, according to the specification of
RDF Compatibility, allows deriving the
triple

a rdf:type D .

Now, the RDF graph S is also an OWL 2 DL ontology. Therefore,
one would expect the triple to be implied according to the
semantics of RIF-OWL DL combinations as well.

To ensure that the RIF-OWL DL combination is faithful to the
OWL 2 DL semantics and to enable using the same, or similar,
RIF rules in combinations with both OWL 2 DL and RDF/OWL 2
Full, the interpretation of frame formulas s[p -> o]
in RIF-OWL DL combinations is slightly different from
their interpretation in RIF and syntactical restrictions are
imposed on the use of variables and function terms in frame
formulas.

The remainder of this section formally defines combinations of
RIF rules with OWL 2 DL and OWL 2 Full ontologies and the
semantics of such combinations. A combination consists of a RIF
document and a set of OWL ontologies. The semantics of
combinations is defined in terms of combined models, which are
pairs of RIF semantic multi-structures and OWL 2 DL, respectively OWL 2 Full
interpretations. The interaction between the structures and interpretations is
defined through a number of conditions. Entailment is defined
as model inclusion, as usual.

4.1 Syntax of RIF-OWL Combinations

Since RDF graphs and OWL 2 Full ontologies cannot be
distinguished, the syntax of RIF-OWL 2 Full combinations is the
same as the syntax of RIF-RDF combinations.

The syntax of OWL ontologies in RIF-OWL DL combinations is
given by the structural specification of OWL 2 and the restrictions on OWL 2 DL ontologies OWL2-Syntax. Certain
restrictions are imposed on the syntax of the RIF rules in
combinations with OWL 2 DL. Specifically, the only terms
allowed in class and property positions in frame formulas are
constant symbols. A DL-frame formula is a frame formula
a[b1 -> c1 ... bn ->
cn] such that n≥1 and for every
bi, with 1≤i≤n, it
holds that bi is a constant symbol and if
bi= rdf:type, then
ci is a constant symbol.

Definition. A condition formula
φ is a DL-condition if every
frame formula in φ is a DL-frame formula.

When clear from the context, RIF-OWL DL-combinations are
referred to simply as combinations.

4.1.1 Safeness Restrictions

In the literature, several restrictions on the use of variables
in combinations of rules and Description Logics have been
identified [Motik05,
Rosati06] for the purpose of decidable
reasoning. This section specifies such safeness restrictions
for RIF-OWL2-DL combinations.

Given a set of OWL 2 DL ontologies O, a variable
?x in a RIF rule QH :- B is
DL-safe if it occurs in an atomic formula in B
that is not of the form s[Ps[P -> o]o] or s[rdf:types[rdf:type ->
A]A], where s,P,o,andAareterms(possiblyincluding?x)andP or A occurs in one of the
ontologies in O. A disjunction-free RIF rule Q (H :- B) is
DL-safe, given O, if every variable that occurs
in H :- B is DL-safe. A disjunction-free RIF rule Q (H :- B)
is weakly DL-safe, given O, if every variable
that occurs in H is DL-safe.

The above definition of
DL-safeness is intended to identify a fragment of RIF-OWL DL
combinations for which reasoning is decidable. This definition
should be considered AT RISK and may become stricter
based on implementation experience.

4.1.2 Datatypes in OWL 2

Compared with RDF and the RIF, OWL 2 uses a slightly extended
notion of datatype.

In the remainder of this section, a datatyped contains, in
addition to the lexical space, value space, and
lexical-to-value mapping, a facet space, which is a
set of pairs of the form (F, v), where F is an IRI
and v is a data value, and a facet-to-value mapping,
which is a mapping from facets to subsets of the value space of
d.

An OWL 2 datatype map D is a datatype map that maps the IRIs of the
datatypes specified in
Section 4 of
[OWL2-Syntax] to the
corresponding datatypes such that the domain of D does not
include rdfs:Literal.

4.2 Semantics of RIF-OWL Combinations

The semantics of RIF-OWL2 DL combinations cannot
straightforwardly extend the semantics of RIF-RDF combinations,
because OWL 2 DL does not extend the RDF semantics. In order
to keep the syntax of the rules uniform between RIF-OWL 2 Full
and RIF-OWL DL combinations, the semantics of RIF frame
formulas is slightly altered in RIF-OWL DL combinations.

The semantics of RIF-OWL 2 Full combinations is a
straightforward extension of the semantics of RIF-RDF
combinations. It is based on the same notion of
common interpretations,
but defines additional notions of satisfiability and
entailment.

4.2.2 OWL DL

The semantics of RIF-OWL DL-combinations is similar in spirit
to the semantics of RIF-RDF combinations. Analogous to
common-RIF-RDF-interpretations, there is the notion of
common-RIF-OWL DL-interpretations, which are pairs of RIF and
OWL 2 DL interpretations, and which define a number of
conditions that relate these interpretations to each other.

4.2.2.1 Modified Semantics for RIF Frame Formulas

The modification of the semantics of RIF frame formulas is
achieved by modifying the mapping function for frame formulas
(Iframe), and leaving the RIF BLD
semantics [RIF-BLD] otherwise
unchanged.

Namely, frame formulas of the form s[rdf:type -> o]
are interpreted as membership of s in the set denoted
by o and frame formulas of the form s[p ->
o], where p is not rdf:type, as
membership of the pair (s, o) in the binary relation
denoted by p.

IC is defined like in RIF semantic structures, with the difference that for constants appearing as attribute bi or value ci in any frame formula a[b1 -> c1 ... bn -> cn] it is not required that IC(bi) ∈ Dind or IC(ci) ∈ Dind and ;

Iframe is a mapping from Dind to total functions of the form SetOfFiniteBags(D × D) → D such that for each pair (u, v) ∈ SetOfFiniteBags(D\Dind × D) it holds that u ≠ IC(rdf:type) if and only if v ∈ Dind;

Dl-semantic multi-structures are defined analogous to RIF-BLD semantic multi-structures [RIF-BLD]. Formally, a dl-semantic multi-structureÎ is a set of dl-semantic structures {J,I; Ii1, Ii2, ...}, where

I and J are dl-semantic structures; and

Ii1, Ii2, etc., are dl-semantic structures adorned with the locators of distinct RIF-BLD formulas (one can think of these adorned structures as locator-structure pairs).

All the structures in Î (adorned and non-adorned) are identical in all respects except for the following:

The mappings JC, IC, ICi1, ICi2, ... may differ on the constants in Const that belong to the rif:local symbol space.

The mappings JV, IV, IVi1, IVi2 may differ.

The notation I(φ), for any other formula or symbol φ, and
the truth valuation function TValÎ are defined
as in BLD semantic structures.

4.2.2.2 Semantics of RIF-OWL DL Combinations

As defined in
[OWL2-Semantics], an
interpretation for a vocabulary V over a datatype map
D is a tuple I=< IR, LV, C, OP, DP, I, DT, LT, FA >,
where

IR is a non-empty set, called the object domain,

LV is a non-empty set, called the data domain, which includes all value spaces of the datatypes in the range of D,

C is a mapping from classes to subsets of IR,

OP is a mapping from object properties to subsets of IR × IR,

DP is a mapping from object properties to subsets of IR × LV,

I is a mapping from individuals into IR,

DT is a mapping from datatypes to subsets of LV,

LT is a mapping from typed literals in V into LV, and

FA is a mapping from IRI, literal pairs to subsets of value spaces in D.

The OWL semantics imposes a number of
further restrictions on the mapping functions to ensure the
interpretation of datatypes, literals, and facets conforms with
the given datatype map D and to define the semantics of
built-in classes and properties (e.g., owl:Thing).
The mappings DT, LT, and FA are essentially given by the
datatype map.

C(c) is the set of all objects k such that Itruth(Iframe'(k)({(IC(rdf:type),IC(<c>))})) = t, for every IRI c in V;

OP(p) is the set of all pairs (k, l) such that Itruth(Iframe'(k)({(IC(<p>), l)})) = t (true), for every IRI p identifying an object property in V;

DP(p) is the set of all pairs (k, l) such that Itruth(Iframe'(k)({(IC(<p>), l)})) = t (true), for every IRI p identifying a data property in V;

I(i) = IC(<i>) for every IRI i identifying an individual in V. ☐

Condition 2 ensures that the relevant parts of the domains of
interpretation are the same. Condition 3 ensures that the
interpretation (extension) of an OWL class identified by an IRI
u corresponds to the interpretation of frames of the
form ?x[rdf:type -> <u>]. Conditions 4 and 5
ensure that the interpretation (extension) of an OWL object or
data property identified by an IRI u corresponds to
the interpretation of frames of the form ?x[<u> ->
?y]. Finally, condition 6 ensures that individual
identifiers in the OWL ontologies and the RIF documents are
interpreted in the same way.

Using the definition of common-RIF-OWL DL-interpretation,
satisfaction, models, and entailment are defined in the usual
way:

Example. In OWL 2 DL, the domains for interpreting
individuals respectively, literals (data values), are disjoint.
The disjointness entails that data values cannot be members of
a class and individuals cannot be members of a datatype.

RIF does not make such distinctions; variable quantification
ranges over the entire domain. So, the same variable may be
assigned to an abstract individual or a concrete data value.
Additionally, RIF constants (e.g., IRIs) denoting individuals
can be written in place of a data value, such as the value of a
data-valued property or in datatype membership statements;
similarly for constants denoting data values. Such statements
cannot be satisfied in any common-RIF-OWL DL-interpretation.
The following example illustrates several such statements.

Consider the datatype xs:string and a RIF-OWL DL
combination consisting of the set containing only the OWL DL
ontology

ex:myiri rdf:type ex:A .

and a RIF document containing the following fact

ex:myiri[rdf:type -> xs:string]

This combination is not
OWLRIF-OWL DL-satisfiable, because
ex:myiri is an individual identifier and S maps individual
identifiers to elements in O, which is disjoint from the
elements in the datatype xs:string.

Consider a RIF-OWL DL combination consisting of the set
containing only the OWL 2 DL ontology

ex:hasChild rdf:type owl:ObjectProperty .

and a RIF document containing the following fact

ex:myiri[ex:hasChild -> "John"]

This combination is not
OWLRIF-OWL DL-satisfiable, because
ex:hasChild is an object property, and values of object
properties may not be concrete data values.

Consider a RIF-OWL DL combination consisting of the OWL DL
ontology

SubClassof(ex:A ex:B)

and a RIF document containing the following rule

Forall ?x (?x[rdf:type -> ex:A])

This combination is not
OWLRIF-OWL DL-satisfiable, because the
rule requires every element, including every concrete data
value, to be a member of the class ex:A. However,
since every OWL interpretation requires every member of
ex:A to be an element of the object domain, concrete
data values cannot be members of the object domain. ☐

5 Importing RDF and OWL in RIF

In the preceding sections, RIF-RDF Combinations and RIF-OWL combinations were defined in an abstract way, as pairs consisting of a RIF document and a set of RDF graphs/OWL ontologies. In addition, different semantics were specified based on the various RDF and OWL entailment regimes.
RIF provides a mechanism for explicitly referring to (importing) RDF graphs from documents and specifying the intended profile (entailment regime) through the use of Import statements.

This section specifies how RIF documents with such import statements must be interpreted.

A RIF document contains a number of Import statements. Unary Import statements are used for importing RIF documents, and the interpretation of these statements is defined in Section 3.5 of [RIF-BLD]. This section defines the interpretation of binary Import statements:

Import(<t1> <p1>)
...
Import(<tn> <pn>)

Here, ti is an absolute IRI referring to an RDF graph to be imported and pi is an absolute IRI denoting the profile to be used for the import.

Profiles are ordered as specified in Section 5.1.1. If several graphs are imported in a document, and these imports specify different profiles, the highest of these profiles is used. For example, if a RIF document R imports an RDF graph S1 with the profile RDF and an RDF graph S2 with the profile OWL Full, the notions of OWLRIF-OWL Full-model, OWLRIF-OWL Full-satisfiability, and OWLRIF-OWL Full-entailment must be used with the combination <R, {S1, S2}>.

5.1 Profiles of Imports

RIF defines specific profiles for the different notions of model, satisfiability and entailment of combinations, as well as one generic profile. The use of a specific profile specifies how a combination should be interpreted. If a specific profile cannot be handled by a receiver, the combination should be rejected. The use of a generic profile implies that a receiver may interpret the combination to the best of its ability.

The use of profiles is not restricted to the profiles specified in this document. Any specific profile that is used with RIF must specify an IRI that identifies it, as well as associated notions of model, satisfiability, and entailment for combinations.

5.1.1 Specific Profiles

The following table lists the specific profiles defined by RIF, the IRIs of these profiles, and the notions of model, satisfiability, and entailment that must be used with the profile.

5.2 Interpretation of Profiles

are all the two-ary import statements in R and the documents imported into R and let Profile be the set of profiles corresponding to the IRIs p1,...,pn.

If pi, 1 ≤ i ≤ n, corresponds to a DL profile and ui refers to an RDF graph that is not the RDF representation of an OWL (2) DL ontology, the document should be rejected.

If ui, 1 ≤ i ≤ n, refers to an RDF graph that uses a typed literal of the form "s"^^rif:iri or "s"^^rdf:PlainLiteral, the document must be rejected.

If Profile contains only specific profiles, then:

If Profile does not have a single highest profile, the document must be rejected.

If Profile contains only DL profiles and R is not a DL-document formula, it must be rejected.

If Profile contains only DL profiles and the RDF graphs referred to by u1,...,un are RDF representations of the OWL 2 ontologies O1,....,On, then the combination C=<R,{O1,....,On}> must be interpreted according to the highest among the profiles in Profile.

Otherwise, the combination C=<R,{S1,....,Sn}>, where S1,....,Sn are the RDF graphs referred to by u1,...,un, must be interpreted according to the highest among the profiles in Profile.

If Profile contains a generic profile, then the combination C=<R,{S1,....,Sn}>, where S1,....,Sn are the RDF graphs referred to by u1,...,un, may be interpreted according to the highest among the specific profiles in Profile, if there is one.

6 Conformance Clauses

We define notions of conformance for RIF-RDF and RIF-OWL combinations. We define these notions both for the RIF Core [RIF-Core] and RIF BLD [RIF-BLD] dialects.

Conformance is described in terms of semantics-preserving transformations between the native syntax of a compliant processor and the XML syntax of RIF Core and BLD.

We say that an RDF graph S is a standard RDF graph if for every triple s p o in S, s is an IRI or blank node, p is an IRI, and o is an IRI, literal, or blank node. A combination < R, S > is standard if every graph in S is standard.

Each RIF processor has sets Τ, of supported datatypes and symbol spaces that include the symbol spaces listed in
[RIF-DTB], and Ε, of supported external terms that include the built-ins listed in [RIF-DTB]. The datatype map of a RIF processor is the smallest datatype map conforming with the set of datatypes in Τ.

Now, let P ∈ {Simple, RDF, RDFS, D, OWL Full} be a specific RDF profile. A RIF-RDF combination C=< R, S > is a BLDΤ,Ε-P combination if R is a BLDΤ,Ε formula and C is a CoreΤ,Ε-P combination if R is a CoreΤ,Ε formula.

A RIF-OWL DL-combination C=<
R, O > is a BLDΤ,Ε-OWL DL combination if R is a BLDΤ,Ε formula and C is a CoreΤ,Ε-OWL DL combination if R is a CoreΤ,Ε formula.

A RIF processor is a conformantBLDΤ,Ε-Pconsumer, for P ∈ {Simple, RDF, RDFS, D, OWL DL, OWL Full}, iff it implements a semantics-preserving mapping, μ, from the set of standard BLDΤ,Ε-P combinations, standard RDF graphs, OWL 2 ontologies, and BLDΤ,Ε formulas to the language L of the processor (μ does not need to be an "onto" mapping) and, in case P ∈ {OWL DL, OWL Full}, its datatype map is an OWL 2 datatype map.

We say that a RIF document R is list-safe if R is safe and it contains no occurrences of rdf:first, rdf:rest, or rdf:nil in rule consequents. An RDF graph S is list-safe if it contains no occurrences of rdf:first or rdf:rest outside of the property positions, it contains no occurrences of rdf:nil outside of triples of the form ... rdf:rest rdf:nil, and there are no two triples s rdf:first o1 . s rdf:first o2 . or s rdf:rest o1 . s rdf:rest o2 . in S, where s, o1, o2 are RDF terms and o1≠o2. A combination < R, S > is list-safe if R is list-safe and the merge of the graphs in S is list-safe.

A RIF processor is a conformantCoreΤ,Ε-Pconsumer, for P ∈ {Simple, RDF, RDFS, D, OWL DL, OWL Full}, iff it implements a semantics-preserving mapping, μ, from the set of standard list-safe CoreΤ,Ε-P combinations, standard RDF graphs, OWL 2 ontologies, and CoreΤ,Ε formulas to the language L of the processor (μ does not need to be an "onto" mapping) and, in case P ∈ {OWL DL, OWL Full}, its datatype map is an OWL 2 datatype map.

Formally, this means that for any pair (φ, ψ), where φ is a BLDΤ,Ε-P combination and ψ is an RDF graph, OWL 2 ontology, or BLDΤ,Ε formula such that φ |=P ψ is defined, φ |=P ψ iff μ(φ) |=L μ(ψ). Here |=P denotes P-entailment and |=L denotes the logical entailment in the language L of the RIF processor.

A RIF processor is a conformantBLDΤ,Ε-Pproducer iff it implements a semantics-preserving mapping, ν, from the language L of the processor to the set of all BLDΤ,Ε formulas, RDF graphs, OWL 2 ontologies, and BLDΤ,Ε-P combinations (ν does not need to be an "onto" mapping).

A RIF processor is a conformantCoreΤ,Ε-Pproducer iff it implements a semantics-preserving mapping, ν, from the language L of the processor to the set of all CoreΤ,Ε formulas, RDF graphs, OWL 2 ontologies, and CoreΤ,Ε-P combinations (ν does not need to be an "onto" mapping).

Formally, this means that for any pair (φ, ψ) of formulas in L such that φ |=L ψ is defined, φ |=L ψ iff ν(φ) |=P ν(ψ). Here |=P denotes P-entailment and |=L denotes the logical entailment in the language L of the RIF processor.

7 Acknowledgements

This document is the product of the Rules Interchange Format (RIF) Working Group (see below),
the members of which deserve recognition for their time and commitment
to RIF. The editors extend special thanks to: Mike Dean, Michael Kifer, Stella Mitchell, Axel Polleres, and Dave Reynolds, for their thorough reviews and insightful discussions;
the working group chairs, Chris Welty and Christian de Sainte-Marie, for their invaluable technical help and inspirational leadership; and W3C staff contact Sandro Hawke, a constant source of ideas, help, and feedback.

The following members of the joint RIF-OWL task force have contributed to the OWL Compatibility section in this document: Mike Dean, Peter F. Patel-Schneider, and Ulrike Sattler.

9 Appendix: Embeddings (Informative)

RIF-RDF combinations can be embedded into RIF documents in a fairly straightforward way, thereby demonstrating how a RIF-compliant translator without native support for RDF can process RIF-RDF combinations.
RIF-OWL combinations cannot be embedded in RIF, in the general case. However, there is a subset of OWL 2 DL, namely the OWL 2 RL profile [OWL2-Profiles], for which RIF-OWL combinations that can be embedded.

Simple, RDF, and RDFS entailment for RIF-RDF combinations are embedded in RIF Core. RIF-OWL 2 RL combinations require reasoning with equality, and thus could not be embedded in RIF Core; they are embedded in RIF BLD.

The embeddings are defined using an embedding function tr that maps symbols, triples, and RDF graphs/OWL ontologies to RIF symbols, statements, and documents, respectively.

To embed consistency checking in RDF(S) and OWL, we use a special 0-ary predicate symbol rif:error, which is assumed not to be used in the RIF documents in the combination.

Besides the namespace prefixes defined in the Overview, the following namespace prefix is used in this appendix: pred refers to the RIF namespace for built-in predicates http://www.w3.org/2007/rif-builtin-predicate# [RIF-DTB].

To facilitate the definition of the embeddings we define the notion of a merge of RIF formulas.

Definition. Let R={R1,...,Rn} be a set of document, group, and rule formulas, such that there are no prefix or base directives, or relative IRIs in R and directive11, ..., directivenm are all the import directives occurring in document formulas in R.
The merge of R, denoted merge(R), is defined as Document(directive11 ... directivenm Group(R*1 ... R*n)), where R*i is obtained from Ri in the following way:

if Ri is a document formula of the form Document(directivei1 ... directiveim Γ), then R*i=Γ and

if Ri is a non-document formula (i.e., fact, rule, or group), then R*i=Ri. ☐

Note that the requirement that no prefix or based directives, or relative IRIs are included in any of the formulas to be merged is not a limitation, since compact IRIs can be rewritten to absolutes IRIs, as can relative IRIs, by exploiting prefix and base directives, and the location of the document.

9.1 Embedding RIF-RDF Combinations

RIF-RDF combinations are embedded by combining the RIF rules with embeddings of the RDF graphs and an axiomatization of Simple, RDF, and RDFS entailment.

The embedding is not defined for combinations that include infinite RDF graphs and for combinations that include RDF graphs with RDF URI references that are not absolute IRIs (see the End note on RDF URI references) or plain literals that are not in the lexical space of the xs:string datatype [XML-Schema2]. Also, the embedding is not defined for RDF lists.

We define a list-free combination as a combination that does not contain any mention of the symbols rdf:first, rdf:rest, or rdf:nil.

In the remainder of this section we first define the embedding of symbols, triples, and graphs, after which we define the axiomatization of Simple, RDF, and RDFS entailment of combinations and, finally, demonstrate faithfulness of the embeddings.

9.1.1 Embedding Symbols

Given a combination C=< R,S>, the function tr maps RDF symbols of a vocabulary V and a set of blank nodes B to RIF symbols, as defined in the following table. It is assumed that the vocabulary V includes all the IRIs and literals used in the RIF documents and condition formulas under consideration.

In the table, the mapping tr' is
an injective function that maps typed literals to new constants in the rif:local symbol space, where a new constant is a constant that is not used in the document or its vicinity (i.e., entailed formula or entailing combination). It "generates" a new constant from a typed literal.

Mapping RDF symbols to RIF.

RDF Symbol

RIF Symbol

Mapping

IRI i in VU

Constant with symbol space rif:iri

tr(i) = <i>

Blank node _:x in B

Variable symbol ?x

tr(_:x) = ?x

Plain literal without a language tag xxx in VPL

Constant with the datatype xs:string

tr("xxx") = "xxx"

Plain literal with a language tag "xxx"@lang in VPL

Constant with the datatype rdf:PlainLiteral

tr("xxx"@lang) = "xxx@lang"^^rdf:PlainLiteral

Well-typed literal "s"^^u in VTL

Constant with the symbol space u

tr("s"^^u) = "s"^^u

Non-well-typed literal "s"^^u in VTL

Local constant s-u' that is not used in C and is obtained from "s"^^u

tr("s"^^u) = tr'("s"^^u)

9.1.2 Embedding Triples and Graphs

This section extends the mapping function tr to triples
and defines two embedding functions for RDF graphs. In one embedding (trR), graphs are embedded as RIF documents and variables (originating from blank nodes) are skolemized, i.e., replaced with new constant symbols. In the other embedding (trQ), graphs are embedded as condition formulas and variables (originating from blank nodes) are existentially quantified. The following sections show how these embeddings can be used for reasoning with combinations.

For skolemization we assume a function sk that takes as argument a formula φ and returns a formula φ' that is obtained from φ by replacing every variable symbol ?x with <new-iri>, where new-iri is a new globally unique IRI, i.e., it does not occur in the graph or its vicinity (i.e., entailing combination or entailed graph/formula).

9.1.3 Embedding Simple Entailment

The semantics of the RDF vocabulary does not need to be axiomatized for Simple entailment. Nonetheless, the connection between RIF class membership and subclass statements and the RDF type and subclass statements needs axiomatization. We define:

RSimple

=

Document( Group(

Forall ?x ?y (?x[rdf:type -> ?y] :- ?x # ?y)

Forall ?x ?y (?x # ?y :- ?x[rdf:type -> ?y])

Forall ?x ?y (?x[rdfs:subClassOf -> ?y] :- ?x ## ?y]) ))

The following theorem shows how checking Simple-entailmentRIF-Simple-entailment of combinations can be reduced to checking entailment of RIF conditions by using the embeddings of RDF graphs defined above.

Proof. We prove both directions through contraposition. We first consider condition formulas (the second part of the theorem), after which we consider graphs (the first part of the theorem).
In the proof we abbreviate merge({R, RSimple, trR(S1), ..., trR(Sn)}) with R'.

(=>) Assume R' does not entail φ. This means there is some semantic multi-structure Î that is a model of R', but not of φ.
Consider the pair (Î, I), where I is the interpretation defined as follows:

IR is Dind,

IP is the set of all k in Dind such that there exist some a, b in Dind and Itruth(Iframe(a)(k,b))=t,

LV is the union of the value spaces of all considered datatypes,

IEXT(k) is the set of all pairs (a, b), with a, b, and k in Dind, such that Itruth(Iframe(a)(k,b))=t,

IS(i) is IC(<i>), for every absolute IRI i in VU, and

IL((s, d)) is IC(tr("s"^^d)), for every typed literal (s, d) in VTL.

Clearly, (Î, I) is a common-RIF-RDF-interpretation: conditions 1-6 in the definition are satisfied by construction of I and conditions 7 and 8 are satisfied by condition 4 and by the fact that Î is a model of RSimple.

Consider a graph Si in {S1,...,Sn}. Let x1,..., xm be the blank nodes in Si and let u1,..., um be the new IRIs that were obtained from the variables ?x1,..., ?xm through the skolemization in trR(Si), i.e., ui=sk(?xi). Now, let A be a mapping from blank nodes to elements in Dind such that A(xj)=IC(uj) for every blank node xj in Si. From the fact that I is a model of trR(Si) and by construction of I it follows that [I+A] satisfiesSi (see Section 1.5 of [RDF-Semantics])), and so I satisfies Si.

We have that Î is a model of R, by assumption. So, (Î, I) satisfies C. Again, by assumption, I is not a model of φ. Therefore, C does not entail φ.

Assume now that R' does not entail trQ(T), which means there is a semantic multi-structure Î that is a model of R', but not of trQ(T). The common-RIF-RDF-interpretation (Î, I) is obtained in the same way as above, and so it satisfies C.

We proceed by contradiction. Assume I satisfies T. This means there is some mapping A from the blank nodes x1,...,xm in T to objects in Dind such that [I+A] satisfies T.
Consider now the semantic multi-structure Î*, which is the same as Î, with the exception of the mapping I*V on the variables ?x1,...,?xm, which is defined as follows: Î*V(?xj)=A(xj) for each blank node xj in S. By construction of I and since [I+A] satisfies T we can conclude that I* is a model of And(tr(t1)... tr(tm)), and so I is a model of trQ(T), violating the assumption that it is not. Therefore, (Î, I) does not satisfy T and C does not entail T.

(<=) Assume C does not Simple-entail φ. This means there is some common-RIF-RDF-interpretation (Î, I) that satisfies C such that I is not a model of φ.

Consider the semantic multi-structure Î', which is like Î, except for the mapping I'C on the new IRIs that were introduced by the skolemization mapping sk(). The mapping of these new IRIs is defined as follows:
For each graph Si in {S1,...,Sn}, let x1,..., xm be the blank nodes in Si and let u1,..., um be the new IRIs that were obtained from the variables ?x1,..., ?xm through the skolemization in trR(Si), i.e., uj=sk(?xj). Now, since I satisfies Si, there must be a mapping A from blank nodes to elements in Dind such that [I+A] satisfies Si. We define I'C(uj)=A(xj) for every blank node xj in Si.

By assumption, Î' is a model of R (recall that Î' differs from Î only on the new IRIs, which are not in R). Clearly, I' is also a model of RSimple, by conditions 7, 8, and 4 in the definition of common-RIF-RDF-interpretation.
From the fact that I satisfies Si and by construction of I' it follows that I' is a model of trR(Si). So, I' is a model of R'. Since I is not a model of φ and φ does not contain any of the new IRIs, I' is not the model of φ. Therefore, R' does not entail φ.

Assume now that C does not entail T, which means there is a common-RIF-RDF-interpretation (Î, I) that satisfies C, but I does not satisfy T. We obtain I' from I in the same way as above, and so it satisfies R'. It can be shown analogous to the (=>) direction that if I' is a model of trQ(T), then there is a blank node mapping A such that [I+A] satisfies T, and thus I satisfies T, violating the assumption that it does not. Therefore, I' is not a model of trQ(T) and thus R' does not entail trQ(T). ☐

Proof. The theorem follows immediately from the previous theorem and the observation that a combination (respectively, RIF document) is satisfiable (respectively, has a model) if and only if it does not entail the condition formula "a"="b". ☐

9.1.4 Embedding RDF Entailment

We axiomatize the semantics of the RDF vocabulary using the following RIF rules.

To finitely embed RDF entailment, we need to consider a subset of the RDF axiomatic triples. Given a combination C, the context of C includes C and its vicinity (i.e., all graphs/formulas considered for entailment checking). The set of RDF finite-axiomatic triples is the smallest set such that:

every RDF axiomatic triple not of the form rdf:_i rdf:type rdf:Property is an RDF finite-axiomatic triple, where i is a positive integer,

one triple rdf:_m rdf:type rdf:Property, for some positive integer m such that rdf:_m does not occur in the context of C, is an RDF finite-axiomatic triple, and

if rdf:_j occurs in the context of C, for some positive integer j, then rdf:_j rdf:type rdf:Property is an RDF finite-axiomatic triple.

We assume that the unary predicate symbols ex:wellxml and ex:illxml are not used in the context of the the given combination.

Here, inconsistencies may occur if non-well-typed XML literals, axiomatized using the ex:illxml predicate, are in the class extension of rdf:XMLLiteral. If this situation occurs, rif:error is derived, which signifies an inconsistency in the combination.

Proof.
In the proof we abbreviate merge({RRDF, R, trR(S1), ..., trR(Sn)}) with R'.
The proof is obtained from the proof of correspondence for Simple entailment in the previous section with the following modifications: (*) in the (=>) direction we additionally need to ensure that I does not satisfy rif:error, extend I to ensure it satisfies the RDF axiomatic triples and show that I is an RDF-interpretation, and (**) in the (<=) direction we need to slightly extend the definition of I' to account for ex:wellxml and ex:illxml, and show that I' is a model of RRDF.

(*) We assume that, for every non-well-typed literal of the form (s, rdf:XMLLiteral) in VTL, IC(tr("s"^^rdf:XMLLiteral)) is not in the value space of any of the considered datatypes and tr("s"^^rdf:XMLLiteral)[rdf:type -> rdf:XMLLiteral] is not satisfied in I. Since C is RDF-satisfiable,RIF-RDF-satisfiable, one can verify that this does not compromise satisfaction of R'. Finally, we may assume, without loss of generality, that I does not satisfy rif:error. See also the proof of the following theorem.

For any positive integer j such that rdf:_j does not occur in the context of C, I and I are extended such that IS(rdf:_j)=IC(rdf:_j)=IC(rdf:_m) (see the definition of finite-axiomatic triples above for the definition of m). Clearly, this does not affect satisfaction of R' or non-satisfaction of φ and trQ(T).

To show that I is an RDF-interpretation, we need to show that I satisfies the RDF axiomatic triples and the RDF semantic conditions.
Satisfaction of the axiomatic triples follows immediately from the inclusion of tr(t) in RRDF for every RDF finite-axiomatic triple t, the fact that I is a model of RRDF, and construction of I. Consider the three RDF semantic conditions:

1

x is
in IP if and only if <x, I(rdf:Property)> is in IEXT(I(rdf:type))

2

If "xxx"^^rdf:XMLLiteral is in V and xxx is a well-typed XML literal string, then

Satisfaction of condition 1 follows from satisfaction of the first rule in RRDF in I and construction of I; specifically, the second bullet in the definition.
Consider a well-typed XML literal "xxx"^^rdf:XMLLiteral. By the definition of satisfaction in RIF BLD, IC("xxx"^^rdf:XMLLiteral) is the XML value of xxx (condition 2a), and is clearly in LV (condition 2b), by definition of I. Condition 2c is satisfied by satisfaction of the second rule in RRDF in I.
Satisfaction of 3a and 3b follows straightforwardly from our assumptions on I. This establishes the fact that I is an RDF-interpretation.

(**) Recall that, by assumption, ex:wellxml and ex:illxml are not used in R. Therefore, changing satisfaction of atomic formulas involving ex:wellxml and ex:illxml does not affect satisfaction of R. We assume that I'C(ex:wellxml)=k and I'C(ex:illxml)=l are distinct unique elements, i.e., no other constants is mapped to k and l.
We define I'F(k) and I'F(l) as follows:
For every typed literal of the form (s, rdf:XMLLiteral) such that I'C(tr(s^^rdf:XMLLiteral))=u, if (s, rdf:XMLLiteral) is well-typed, Itruth(I'F(k)(u))=t and Itruth(I'F(l)(u))=f, otherwise Itruth(I'F(k)(u))=f and Itruth(I'F(l)(u))=t; I'truth(I'F(k)(v))=Itruth(I'F(l)(v))=f for every other object v in Dind.

Consider RRDF. Satisfaction of RSimple was established in the proof in the previous section. Satisfaction of the facts corresponding to the RDF axiomatic triples in I' follows immediately from the definition of common-RIF-RDF-interpretation and the fact that I is an RDF-interpretation, and thus satisfies all RDF axiomatic triples.
Satisfaction of the ex:wellxml and ex:illxml facts in RRDF follows immediately from the definition of I'. Finally, satisfaction of the rules in RRDF follow straightforwardly from the RDF semantic conditions 1, 2, and 3. This establishes the fact that I' is a model of RRDF. ☐

Proof. Recall that we assume rif:error does not occur in R. If <R,{S1,...,Sn}> is not RDF-satisfiable,RIF-RDF-satisfiable, then either merge({R, trR(S1), ..., trR(Sn)}) is not consistent, or condition 3a or 3b (see previous proof) is violated. In either case, rif:error is entailed.
If rif:error is entailed, either merge({RRDF, R, trR(S1), ..., trR(Sn)}) is inconsistent, which means merge({R, trR(S1), ..., trR(Sn)}) is not consistent and thus <R,{S1,...,Sn}> is not RDF-satisfiable,RIF-RDF-satisfiable, or the body of the second or third rule in RRDF is satisfied in every model, which means either condition 3a or 3b is violated, and so <R,{S1,...,Sn}> is not RDF-satisfiable.RIF-RDF-satisfiable. ☐

9.1.5 Embedding RDFS Entailment

We axiomatize the semantics of the RDF(S) vocabulary using the following RIF rules.

Similar to the RDF case, the set of RDFS finite-axiomatic triples is the smallest set such that:

every RDFS axiomatic triple not of the form rdf:_i ... ..., where i is a positive integer, is an RDFS finite-axiomatic triple,

the triples rdf:_m rdf:type rdfs:ContainerMembershipProperty, rdf:_m rdfs:domain rdfs:Resource, and rdf:_m rdfs:range rdfs:Resource, for some positive integer m such that rdf:_m does not occur in the context of C, are RDFS finite-axiomatic triples, and

if rdf:_j occurs in the context of the combination C, for some positive integer j, then rdf:_j rdf:type rdfs:ContainerMembershipProperty, rdf:_j rdfs:domain rdfs:Resource, and rdf:_j rdfs:range rdfs:Resource are RDFS finite-axiomatic triples.

We assume that the unary predicate symbol ex:welllit is not used in the context of the given combination.

In the following theorems it is assumed that, in combinations C=<R,{S1,...,Sn}>, R does not have mentions of rdfs:Resource, S1,...,Sn do not have mentions of rdfs:Resource beyond triples of the form xxx rdf:type rdfs:Resource, and entailed graphs T and formulas φ do not have mentions of rdfs:Resource.

9.2 Embedding RIF-OWL 2 RL Combinations

It is known that expressive Description Logic languages such as OWL 2 DL cannot be straightforwardly embedded into typical rules languages such as RIF BLD [RIF-BLD], because of features such as disjunction and negation.

In this section we consider a subset of OWL 2 DL in RIF-OWL DL combinations, namely, the OWL 2 RL profile [OWL2-Profiles], and show how reasoning with RIF-OWL 2 RL combinations can be reduced to reasoning with RIF.

The embedding of RIF-OWL 2 RL combinations is not defined for combinations that include infinite OWL ontologies and for combinations that include ontologies with RDF URI references that are not absolute IRIs or plain literals that are not in the lexical space of the xs:string datatype. In addition, the guard predicate pred:is-dt [DTB] must be defined for each datatype used in the combination, where dt is the short name of the datatype.

Since OWL 2 RL includes equality through ObjectMaxCardinality and DataMaxCardinality restrictions, as well as FunctionalObjectPropertyUniverseFunctionalObjectProperty, SameIndividual, and HasKey axioms, and there is non-trivial interaction between such equality and the predicates in the RIF rules in the combination, embedding RIF-OWL 2 RL combinations into RIF requires equality. Therefore, the embedding presented in this appendix is not in RIF Core, even if the RIF document in the combination is. If the ontologies in the combination do not contain any of the mentioned constructs, the embedding is in Core. Also, it is well-known that adding equality to a rules language does not increase its expressiveness in the absence of function symbols: one can replace equality = with a new binary predicate symbol, and add rules for reflexivity and the principle of substitutivity (also called the replacement property).

9.2.1 Embedding RIF DL-document formulas into RIF BLD

Recall that the semantics of frame formulas in DL-document formulas is different from the semantics of frame formulas in RIF documents.
Nonetheless, DL-document formulas can be embedded into RIF documents, by translating frame formulas to predicate formulas.
The mapping tr is the identity mapping on all RIF formulas, with the exception of frame formulas, as defined in the following table.

In the table, the mapping tr' is
an injective function that maps constants to new constants, i.e., constants that are not used in the original document or its vicinity (i.e., entailed or entailing formula). It "generates" a new constant from an existing one.

Proof. We prove both directions by contradiction: if the entailment does not hold on the one side, we show that it also does not hold on the other.

(=>) Assume tr(R) does not entail tr(φ). This means there is some semantic multi-structure I = <TV, DTS, D, Dind,
Dfunc, IC, IV,
IF, Iframe, ISF,
Isub, Iisa, I=,
Iexternal, Itruth> that is a model of tr(R), but not of tr(φ).
Consider the dl-semantic multi-structure I* = <TV, DTS, D, Dind,
Dfunc, I*C, IV,
IF, I*frame', ISF,
Isub, Iisa, I=,
Iexternal, Itruth>, with I*C and I*frame' defined as follows: Let t be an element in D such that Itruth(t)=t and let f in D be such that Itruth(f)=f.

for every constant c' used as unary or binary predicate symbol in tr(R) or tr(φ) such that c'=tr'(c) for some constant c, I*C( c')=IC(c); I*C(c*)=I*C(c*) for every other constant c*;

for every constant c' used as unary predicate symbol in tr(R) or tr(φ) such that c'=tr'(c) for some constant c, and every object k in Dind, if Itruth(IF(IC(c'))(k))=t, I*frame'(k)((IC(rdf:type), IC(c))=t,

for every constant b' used as binary predicate symbol in tr(R) or tr(φ) such that b'=tr'(b) for some constant b, and every pair (k, l) in Dind × Dind, if Itruth(IF(IC(b'))(k,l))=t, I*frame'(k)((IC(b),l))=t,

if I*frame'(k)((b1,...,bn))=t and I*frame'(k)((c1,...,cm))=t for any two finite bags (b1,...,bn) and (c1,...,cm), then I*frame'(k)((b1,...,bn,c1,...,cm))=t, and

I*frame'(b)=f for any other bag b.

Observe that tr(R) and tr(φ) do not include frame formulas.
To show that I* is a model of R and not of φ, we only need to show that (+) for any frame formula a[b -> c] that is a DL-condition, I* is a model of a[b -> c] iff I is a model of tr(a[b -> c]). This argument straightforwardly extends to the case of frames with multiple bis and cis, since in RIF semantic structures the following condition is required to hold: TValI(a[b1->c1 ... bn->cn]) = t if and only if TValI(a[b1->c1]) = ... = TValI(a[bn->cn]) = t [RIF-BLD].

Consider the case b=rdf:type. Then,
I* is a model of a[b -> c] iff Itruth(I*frame'(I(a))(IC(rdf:type),IC(c)))=t.
From the definition of I* we obtain
Itruth(I*frame'(I(a))(IC(rdf:type),IC(c)))=t iff I*frame'(I(a))(IC(rdf:type),IC(c))=t.
By definition of the embedding, we know that tr'(c) is used as unary predicate symbol in tr(R) or tr(φ). From the definition of I* we obtain
I*frame'(I(a))(IC(rdf:type),IC(c))=t iff Itruth(IF(IC(tr'(c)))(I(a)))=t.
Finally, since tr(a[b -> c])=tr'(c)(a), we obtain
Itruth(IF(IC(tr'(c)))(I(a)))=t iff I is a model of tr(a[b -> c]).
From this chain of equivalences follows that I* is a model of a[b -> c] iff I is a model of tr(a[b -> c]).

The argument for the case b≠rdf:type is analogous, thereby obtaining (+).

(<=) Assume R does not dl-entail φ. This means there is some dl-semantic multi-structure I = <TV, DTS, D, Dind,
Dfunc, IC, IV,
IF, Iframe', ISF,
Isub, Iisa, I=,
Iexternal, Itruth> that is a model of R, but not of φ. Let B be the set of constant symbols occurring in the frame formulas of the forms a[rdf:type -> b] and a[b -> c] in R or φ.
Consider the semantic multi-structure I* = <TV, DTS, D, Dind,
Dfunc, I*C, IV,
I*F, I*frame, ISF,
Isub, Iisa, I=,
Iexternal, Itruth>. Let t and f in D be such that Itruth(t)=t and Itruth(f)=f. We define I*C, I*frame, and I*F as follows:

I*C(tr'(b))=I*C(b) for any b in B ; I*C(c)=IC(c) for any c not in B,

I*frame(b)=f for any finite bag b of D, and

I*F is defined as follows:

for every c in B, given an object k in Dind, if Itruth(Iframe'(k)((IC(rdf:type), IC(c)))=t, I*F(I*C(tr'(c)))(k)=t; I*F(I*C(tr'(c)))(k')=f for any other k' in Dind,

for every b in B, given a pair (k, l) in Dind × Dind, if Itruth(Iframe(k)((IC(b),l)))=t, I*F(tr'(b))(k,l)=t;I*F(tr'(b))(k',l')=f for any other pair (k', l') in Dind × Dind, and

I*F(c')=IF(c') for every other constant c'.

Observe that R and φ do not include predicate formulas involving derived constant symbols tr'(b) or tr'(c). The remainder of the proof is analogous to the (=>) direction.
☐

9.2.2 Embedding OWL 2 RL into RIF BLD

The embedding of OWL 2 RL into RIF BLD has two stages: normalization and embedding.

The OWL 2 syntax is given in terms of a structural model, and there is a functional-style syntax that is a serialization of this structural model. For convenience, normalization and embedding in this section are done in terms of the functional-style syntax. That is, the normalization mapping takes as input a functional-style syntax ontology document and produces a normalized ontology document. The embedding mapping takes as input a normalized ontology document and produces an RIF document. We refer to Section 4.2 of [OWL2-Profiles] for the specification of the OWL 2 RL syntax.

9.2.2.1 Normalization of OWL 2 RL

Normalization splits the OWL axioms so that the later mapping to RIF of the individual axioms results in rules. Additionally, it simplifies the axioms and removes annotations.

It is assumed that the normalization process is preceded by a simplification process that removes all namespace prefixes, turns all CURIEs and relative IRIs into absolute IRIs, and removes all annotations, import statements, entity declarations, and annotation axioms.

We note here that, strictly speaking, simplified OWL 2 RL ontologies are not OWL 2 RL ontologies in the general case, because certain entity declarations are required (e.g., those distinguishing data from object properties). It is assumed that such entity declarations are present implicitly, i.e., they do not appear explicitly in the simplified ontology, but they are known.
We also note that removing import statements in the simplification does not prohibit importing ontologies in practice; since combinations contain sets of ontologies, all imported ontologies may be added to these sets.
The normalization mapping trN takes as input a simplified ontology O and produces an equivalent normalized ontology O'.

The names of variables used in the mapping generally correspond to the names of productions in the OWL 2 RL grammar.

9.2.2.2 Embedding Normalized OWL 2 RL

We now proceed with the embedding of normalized OWL 2 RL ontologies into RIF DL-document formulas. The embedding function trO takes as input a normalized OWL 2 RL ontology and returns a RIF-BLD DL-document formula. The embeddings of IRIs and literals is as defined in the Section Embedding Symbols.

When we speak about class IDs and datatype IDs we mean IRIs used as identifiers for classes, respectively datatypes. By the syntactic restrictions on OWL ontologies, these sets are disjoint. Similarly, we speak about data, object, and annotation property IDs when talking about IRIs used as identifiers for data, object, respectively annotation properties. These sets of property identifiers are mutually disjoint for any OWL ontology.

Besides the embedding in the previous table, we also need an axiomatization of some of the aspects of the OWL DL semantics, e.g., separation between individual and datatype domains. This axiomatization is defined relative to an OWL vocabulary V, which includes all well-typed literals used in the rules, and a datatype map D, which includes all considered datatypes. In the table, for a given datatype d, L2V(d) is the lexical-to-value mapping of d.

Theorem Given a datatype map D conforming with T, a RIF-OWL-DL-combination <R,{O1,...,On}>, where {O1,...,On} is an imports-closed set of OWL 2 RL ontologies with vocabulary V, is owl-dl-satisfiable with respect to D iff tr(merge({R, ROWL-DL(V), trO(trN(O1)), ..., trO(trN(On))}) does not entail rif:error.

10 Appendix: Change log (Informative)

ErrorIn the definition of dl-semantic-structurecommon-RIF-RDF-interpretation,"RDFinterpretation" was corrected;thepreviousversionofthedefinitiondidnotdisallowclassandpropertyidentifierstobemappedintheindividualdomain.SomemissingCURIEprefixeswereaddedintheAppendix,referencestorif:texthavebeenupdatedtordf:PlainLiteral,inlinereplaced with "Simpleinterpretation". The changein[RIF-DTB],andsometyposwerecorrectedinthemaintext.TheembeddingsoftheSimple,RDF,andRDFSentailmentregimeshavebeenupdatedsothattheyarenowinRIFCore.Itoriginalreferenceto"RDFinterpretation" was previouslyundefinedwhathappenswhenimportinggraphswiththeOWLDLprofile,butthedocumentisnotaDL-documentformulaoranerror.

11 End Notes

RDF URI References: There are certain RDF URI references that are not IRIs (e.g., those containing spaces). It is possible to use such RDF URI references in RDF graphs that are combined with RIF rules. However, such URI references cannot be represented in RIF rules and their use in RDF is discouraged.

Generalized RDF graphs: Standard RDF graphs, as defined in [RDF-Concepts], do not allow the use of literals in subject and predicate positions and blank nodes in predicate positions. The RDF Core working group has listed two issues questioning the restrictions that literals may not occur in subject and blank nodes may not occur in predicate positions in triples. Anticipating lifting of these restrictions in a possible future version of RDF, we use the more liberal notion of generalized RDF graph. We note that the definitions of interpretations, models, and entailment in the RDF semantics document [RDF-Semantics] also apply to such generalized RDF graphs.

We note that every standard RDF graph is a generalized RDF graph. Therefore, our definition of combinations applies to standard RDF graphs as well.

We note also that the notion of generalized RDF graphs is more liberal than the notion of RDF graphs used by SPARQL; generalized RDF graphs additionally allow blank nodes and literals in predicate positions.